Convexity for nabla and delta fractional differences

Abstract

In this paper we demonstrate that

Theorem A

If $\mathrm{f:}\mathbb{N}{}_{a+1}\to \mathbb{R}$ satisfies ${\displaystyle {\nabla }_a^{\nu }}f\left(t\right)\ge 0$, for each $t\in \mathbb{N}{}_{a+1}$, with $2<\nu <3$, then $\nabla {}^{2}f\left(t\right)\ge 0$, for $t\in \mathbb{N}{}_{a+3}$.

Theorem B

If $f:\mathbb{N}{}_a\to \mathbb{R}$ satisfies ${\displaystyle {\Delta }_a^{\nu }}f\left(t\right)\ge 0$, for each $t\in \mathbb{N}{}_a$, with $2<\nu <3$, and $f\left(a\right)\le 0,\Delta f\left(a\right)\ge 0,\Delta {}^{2}f\left(a\right)\ge 0$. Then $\Delta f\left(t\right)\ge 0$, for $t\in \mathbb{N}{}_a$.

This demonstrates that, in some sense, the positivity of the $\nu$th order fractional difference has a strong connection to the convexity of $f\left(t\right)$.

In Section 3, by means of an example, we show that a recent result in {C. Goodrich, A convexity result for fractional differences, Appl. Math. Lett. 35 (2014), pp. 58–62.} is incorrect as stated.

Keywords::

• nabla fractional difference
• convexity
• Taylor monomial
• difference equations

AMS Subject Classification::

• 39A12
• 39A70

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

^1. Email: [email protected]

^2. Email: [email protected]

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 11271380] and Guangdong Province Key Laboratory of Computational Science.